Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

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Abstract

In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

Abstract

In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given.

1 Introduction

Fractional differential equations (FDEs) have become under remarkable consideration as being the generalisation form of the differential equations of integer order and due to their roles in the modelling of several physical processes. Many nonlinear phenomena in physics can be described via FDEs such as chemical physics, optical fibers, plasma, electromagnetic waves, diffusion processes, vibrations in a nonlinear string and etc [1, 2, 3, 4, 5]. Seeking different types of solutions to FDEs for such phenomena has become the subject of interest for researchers. Thus, a lot of powerful mathematical methods have been applied to obtain exact analytic solutions of FDEs, namely, the extended tanh-function method [6, 7], the exp-function method [8, 9], the sub-equation method [10, 11], the improved tan(ϕ/2)-expansion method [12, 13], the (G′/G)-expansion method [14, 15], the modified trial equation method [16, 17], the new extended direct algebraic method [18, 19], the extended sinh-Gordon equation expansion method [20, 21], the unified method [22] and so on.

One of the important physical model is the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation [23, 24, 25] of the form

Dtαu+δu2Dxαu+Dx3αu+DxαDy2αu+DxαDz2αu=0,

where 0 < α ≤ 1 and δ is an arbitrary constant. This model is developed for a plasma comprised of cool and hot electrons and a species of fluid ions [26]. Recently, Sahoo and Ray [10] have applied the improved fractional sub equation method with the aid of Jumarie’s modified Riemann–Liouville derivative for space-time fractional mKdV–ZK equation and they derived exact solutions. Then, the fractional derivatives in the sense of modified Riemann–Liouville derivative and three different solution methods have been employed by Guner et. al. [27] to obtain a variety of exact solutions to equation (1). Moreover, using the ansatz method and the functional variable method, Guner [28] examined exact analytic solutions for equation (1) and constructed singular soliton solutions. In the two latter studies, the fractional complex transform method is used to reduce equation (1) into ordinary differential equation (ODE). However, in a recent study, this transformation has been proved to be incorrect by Herzallah [29]. Subsequently, all obtained solutions by means of this method are wrong.

The present study focuses on investigating the exact analytic solutions to space-time fractional mKdV–ZK equation using the variable separated ODE method [30, 31]. The structure of this work is organised as follows. In Section 2, we introduce the definition of conformable fractional derivative [32] and its properties which will be utilised to reduce FDE into an ODE. Section 3 contains the description of variable separated ODE method and the technique of implementing it to ODEs. In Section 4, the proposed method will be applied to construct the solitons and periodic wave solutions of space-time fractional mKdV–ZK equation. Then, the behaviour of some obtained solutions is displayed graphically. Finally, our discussions and conclusions are presented in Section 5.

2 Conformable fractional derivative

Khalil, et. al. [32] introduced a completely new definition of fractional calculus with the limit operator which is more natural and effective on satisfying some conventional properties than the existing fractional derivatives.

The definition of conformable fractional derivative is given as follows:

Definition 1

Let f: (0, ∞) ⟶ ℝ, then the conformable fractional derivative of f of order α is defined as

Dtαf(t)=limε0f(t+εt1α)f(t)ε

for all t > 0, α ∈ (0, 1).

It is said that if the conformable fractional derivative of f of order α exists, then f is α-differentiable. The conformable fractional derivative satisfies the properties shown in the following theorems:

Theorem 1

Let α ∈ (0, 1] and f = f(t), g = g(t) be α-differentiable at a point t > 0, then:

  1. Dtα(af+bg)=aDtαf+bDtαg, for all a, b ∈ ℝ.
  2. Dtα(tμ)=μtμα, for all μ ∈ ℝ.
  3. Dtα(fg)=fDtαg+gDtαf.
  4. Dtαfg=gDtαffDtαgg2.

Moreover, if f is differentiable, then Dtα(f(t))=t1αdfdt.

Theorem 2

Let f: (0, ∞) ⟶ ℝ be a function such that f is differentiable and also α-differentiable. Let g be a function defined in the range of f and also differentiable; then, one has the following rule:

Dtα(fg)(t)=t1αg(t)f(g(t)),

where prime denotes the classical derivatives with respect to t.

Remark 1

We may use the notation αtαfforDtα(f(t)) to denote the conformable fractional derivatives of f with respect to the variable t of order α.

3 The variable separated ODE method

In this section we shall describe the technique of implementing the proposed method to FDEs in order to extract exact analytic solutions. Suppose that a nonlinear conformable fractional partial differential equation, say, in four independent variables x, y, z and t is given by

P(u,αutα,αuxα,αuyα,αuzα,2αut2α,2αux2α,2αuy2α,2αuz2α,)=0,

where u(x, y, z, t) is an unknown function, P is a polynomial in u and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved. This method consists of the following steps:

  1. Step 1Using the wave transformation
    u(x,y,z,t)=ϕ(ξ),ξ=k1xαα+k2yαα+k3zαα+νtαα,
    where k1, k2, k3 and ν are constants to be determined later, one can find
    αutα=νdϕdξ,αuxα=k1dϕdξ,αuyα=k2dϕdξ,αuzα=k3dϕdξ,2αut2α=ν2d2ϕdξ2,
    Employing (5) and (6), equation (4) is reduced into the following ODE
    P(ϕ,νdϕdξ,k1dϕdξ,k2dϕdξ,k3dϕdξ,ν2d2ϕdξ2,k12d2ϕdξ2,k22d2ϕdξ2,k32d2ϕdξ2,)=0.
  2. Step 2Assuming that equation (7) has the solution of the form
    ϕ(ξ)=j=NNaj(d+F(ξ))j,
    where aN or aN might be zero, but both of them could not be zero simultaneously. The coefficients aj (j = 0, ±1, ±2, …, ± N) and d are constants to be determined, whereas F(ξ) satisfies the general elliptic equation of the form
    F2(ξ)=c0+c1F(ξ)+c2F2(ξ)+c3F3(ξ)+c4F4(ξ),
    where ci (i = 0, 1, 2, 3, 4) are constants. Recently, Sirendaoreji [31] presented five classifications of solutions for equation (9) in accordance with the presence of coefficients ci (i = 0, 1, 2, 3, 4) and they are given as: (1) c0 = c1 = 0, (2) c3 = c4 = 0, (3) c1 = c3 = 0, (4) c2 = c4 = 0, and (5) c0 = 0. Herein, we only concentrate on two cases which are Case I c0 = c1 = 0 and Case II c1 = c3 = 0, for convenience. Solution structures of both cases are exhibited as follows.
    1. Case Ic0 = c1 = 0:
      F1(ξ)=2c2εΔcosh(c2ξ)c3,Δ>0,c2>0,
      F2(ξ)=2c2εΔsinh(c2ξ)c3,Δ<0,c2>0,
      F3(ξ)=2c2εΔcos(c2ξ)c3,Δ>0,c2<0,
      F4(ξ)=2c2εΔsin(c2ξ)c3,Δ>0,c2<0,
      F5(ξ)=c2c31+εtanhc22ξ,Δ=0,c2>0,
      F6(ξ)=c2c31+εcothc22ξ,Δ=0,c2>0,
      F7(ξ)=εc4ξ,c2=c3=0,c4>0,
      F8(ξ)=4c3c32ξ24c4,c2=0,
      where Δ=c324c2c4, ε = ±1.
    2. Case IIc1 = c3 = 0:
      F9(ξ)=εc22c4tanhc22ξ,Δ1=0,c2<0,c4>0,
      F10(ξ)=εc22c4cothc22ξ,Δ1=0,c2<0,c4>0,
      F11(ξ)=εc22c4tanc22ξ,Δ1=0,c2>0,c4>0,
      F12(ξ)=εc22c4cotc22ξ,Δ1=0,c2>0,c4>0,
      F13(ξ)=c2m2c4(m2+1)snc2m2+1ξ,c0=c22m2c4(m2+1)2,c2<0,c4>0,
      F14(ξ)=c2m2c4(2m21)cnc22m21ξ,c0=c22m2(m21)c4(2m21)2,c2>0,c4<0,
      F15(ξ)=c2c4(2m2)dnc22m2ξ,c0=c22(1m2)c4(2m2)2,c2>0,c4<0,
      F16(ξ)=m222c2c4(2m2)sn2c22m2ξ1±dn2c22m2ξ,c0=c22m44c4(2m2)2,c2<0,c4>0,
      F17(ξ)=122c2c4(12m2)sn2c212m2ξ1±cn2c212m2ξ,c0=c224c4(12m2)2,c2>0,c4>0,
      F18(ξ)=ε4c0c414ds(4c0c4)14ξ,22,c2=0,c0c4<0,
      F19(ξ)=εc0c414ns2(c0c4)14ξ,22+cs2(c0c4)14ξ,22,c2=0,c0c4>0,
      where Δ1=c224c0c4, ε = ±1. Here, sn(ξ) = sn(ξ, m), cn(ξ) = cn(ξ, m), dn(ξ) = dn(ξ, m) are called the Jacobian elliptic sine function, the Jacobian elliptic cosine function and the Jacobian elliptic function of the third kind and 0 < m < 1 is the modulus of Jacobian elliptic function. The other Jacobian functions are generated by these three kinds of functions as follows:
      ns(ξ)=1/sn(ξ),nc(ξ)=1/cn(ξ),nd(ξ)=1/dn(ξ),sc(ξ)=sn(ξ)/cn(ξ),sd(ξ)=sn(ξ)/dn(ξ),cd(ξ)=cn(ξ)/dn(ξ),cs(ξ)=cn(ξ)/sn(ξ),ds(ξ)=dn(ξ)/sn(ξ),dc(ξ)=dn(ξ)/cn(ξ).
    3. Step 3The value of the positive integer N can be determined by balancing the highest order linear terms with the nonlinear terms of the highest order emerging in equation (7).
    4. Step 4Substituting (8) and (9) into equation (7), we collect all terms with the same power of (d + F(ξ)). Equating each coefficient of the resulting polynomial to zero, yields a set of algebraic equations for aj (j = 0, ±1, ±2, …, ± N), ci (i = 0, 1, 2, 3, 4), kl (l = 1, 2, 3), d and ν.
    5. Step 5Substituting the values of the constants obtained by solving the algebraic equations extracted in Step 4 together with the solutions of equation (9) into (8), we arrive at different types of solutions for equation (4).

4 Application of the method to the (3 + 1)-dimensional space–time fractional mKdV–ZK equation

Now, we aim to solve the fractional mKdV–ZK equation (1) by applying the variable separated ODE method described above. Thus, to deal with the complex form of equation (1) we will reduce it to an ODE using the transformation (5). The substitution of the transformation (5) into equation (1) leads to

νϕ+δk1ϕ2ϕ+k1(k12+k22+k32)ϕ=0,

where prime denotes the derivative with respect to ξ. Integrating equation (31) once with respect to ξ, we obtain

νϕ+Aϕ3+Bϕ=0,

where A = δk1/3, B=k1(k12+k22+k32) and the integration constant is taken to be zero. Now, we assume that equation (32) has a solution in the form of equation (8). The homogeneous balance between the term ϕ″ and the term ϕ3 in equation (32) gives rise to N = 1. Hence, the solution to equation (32) is given in the form

ϕ(ξ)=a1(d+F(ξ))+a0+a1(d+F(ξ)),

where a−1, a0, a1 are unknown constants to be computed. Substituting equation (33) into equation (32) and using equation (9), we obtain polynomials in (d + F(ξ))j and (d + F(ξ))j, (j = 0, 1, 2, 3). Collecting all coefficients of identical power of the resulting polynomials and equating each coefficient to zero, yield the following set of algebraic equations

B(2a1c4d42a1c3d3+2a1c2d22a1c1d+2a1c0)+Aa13=0,

B(6a1c4d3+92a1c3d23a1c2d+32a1c1)+3Aa12a0=0,

B(6a1c4d23a1c3d+a1c2)+A(3a12a1+3a1a02)+νa1=0,

B(2a1c4d+12a1c3a1c2d+32a1c3d22a1c4d3+12a1c1)+A(6a1a0a1+a03)+νa0=0,

B(3a1c3d+6a1c4d2+a1c2)+A(3a1a12+3a02a1)+νa1=0,

B(6a1c4d+32a1c3)+3Aa0a12=0,

2Ba1c4+Aa13=0.

Solving equations (34)(40) gives the following cases of solutions.

  1. Case IIn this case, c0 = c1 = 0. This case has seven different sets of coefficients for the solution of equation (33) displayed as follows.
    1. Set Ia0=d2c4BA,a1=2c4BA, a−1 = 0, ν = −c2 B, c3 = 0
      u(x,y,z,t)=ε6c2(k12+k22+k32)δsechc2ξ,
      where c2 > 0 and δ > 0.
      u(x,y,z,t)=ε6c2(k12+k22+k32)δcschc2ξ,
      where c2 > 0 and δ < 0.
      u(x,y,z,t)=ε6c2(k12+k22+k32)δsecc2ξ,
      u(x,y,z,t)=ε6c2(k12+k22+k32)δcscc2ξ,
      where c2 < 0 and δ < 0. In solutions (41)(44), ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα.
    2. Set IIa0=(c34c4d)4c42c4BA,a1=2c4BA,a1=0,ν=c2B2,c4=c324c2
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δtanh12c2ξ,
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δcoth12c2ξ,
      where c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+c2k12(k12+k22+k32)tαα.
    3. Set IIIa0=(c34c4d)4c42c4BA,a1=0,a1=d(c32c4d)2c42c4BA,ν=c2B2,c4=c324c2
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δc2+(c2c3d)tanh12c2ξ(c2c3d)+c2tanh12c2ξ,
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δc2+(c2c3d)coth12c2ξ(c2c3d)+c2coth12c2ξ,
      where c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+c2k12(k12+k22+k32)tαα.
    4. Set IVa0=2BA(c2c4d2),a1=0,a1=d2BA(c2c4d2),ν=c2B,c3=2c2dThis gives us the solutions (41)(44).
    5. Set Va0=0,a1=2c4BA,a1=d22c4BA,ν=2c2B,c3=c2d,c4=c24d2
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δ1+tanh212c2ξtanh12c2ξ,
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δ1+coth212c2ξcoth12c2ξ,
      where c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+2c2k1(k12+k22+k32)tαα.
    6. Set VIa0 = 0, a1=2c4BA,a1=d22c4BA,ν=c2B,c3=c2d,c4=c24d2
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δsech212c2ξtanh12c2ξ,
      u(x,y,z,t)=12ε6c2(k12+k22+k32)δcsch212c2ξcoth12c2ξ,
      where c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα.
    7. Set VIIa0=2d2c4BA,a1=2c4BA,a1=d22c4BA,ν=4c2B,c3=c2d,c4=c24d2
      u(x,y,z,t)=2ε6c2(k12+k22+k32)δ12cosh2c2ξ1,
      where c2 > 0 and δ > 0.
      u(x,y,z,t)=2ε6c2(k12+k22+k32)δ12cos2c2ξ1,
      where c2 < 0 and δ < 0. In solutions (53) and (54), ξ=k1xαα+k2yαα+k3zαα4c2k1(k12+k22+k32)tαα.
  2. Case IIIn the second case, c1 = c3 = 0. This case has five different sets of coefficients for the solution of equation (33) shown as follows.
    1. Set Ia0=d2c4BA,a1=2c4BA,a1=0,ν=c2B
      u(x,y,z,t)=εm6c2(k12+k22+k32)δ(m2+1)snc2m2+1ξ,
      where c0=c22m2c4(m2+1)2, c2 < 0 and δ < 0. As m → 1, solution (55) reduces to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δtanhc22ξ.
      u(x,y,z,t)=εm6c2(k12+k22+k32)δ(2m21)cnc22m21ξ,
      where c0=c22m2(m21)c4(2m21)2, c2 > 0 and δ > 0. As m → 1, solution (57) reduces to solution (41).
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ(2m2)dnc22m2ξ,
      where c0=c22(1m2)c4(2m2)2, c2 > 0 and δ > 0. As m → 1, solution (58) reduces to solution (41).
      u(x,y,z,t)=εm23c2(k12+k22+k32)δ(2m2)sn2c22m2ξ1+εdn2c22m2ξ,
      where c0=c22m44c4(2m2)2, c2 < 0 and δ < 0. As m → 1, solution (59) becomes
      u(x,y,z,t)=ε3c2(k12+k22+k32)δtanh2c2ξ1+εsech2c2ξ.
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ(12m2)sn2c212m2ξ1+εcn2c212m2ξ,
      where c0=c224c4(12m2)2, c2 > 0 and δ < 0. As m → 0, solution (61) degenerates to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δsin2c2ξ1+εcos2c2ξ,
      and as m → 1, solution (61) reduces to solution (60). In solutions (55)(62), ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα.
    2. Set IIa0 = 0, a1 = 0, a1=2c0BA, ν = − c2 B, d = 0
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ(m2+1)nsc2m2+1ξ,
      where c0=c22m2c4(m2+1)2, c2 < 0 and δ < 0. As m → 0, solution (63) degenerates to solution (44) while as m → 1, solution (63) reduces to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δcothc22ξ.
      u(x,y,z,t)=ε6c2(m21)(k12+k22+k32)δ(2m21)ncc22m21ξ,
      where c0=c22m2(m21)c4(2m21)2, c2 > 0 and δ > 0. As m → 0, solution (65) degenerates to solution (43).
      u(x,y,z,t)=ε6c2(1m2)(k12+k22+k32)δ(2m2)ndc22m2ξ,
      where c0=c22(1m2)c4(2m2)2, c2 > 0 and δ > 0.
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ(2m2)1+εdn2c22m2ξsn2c22m2ξ,
      where c0=c22m44c4(2m2)2, c2 < 0 and δ < 0. As m → 0, solution (67) gives rise to solution (44) and as m → 1, solution (67) changes to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ1+εsech2c2ξtanh2c2ξ.
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ(12m2)1+εcn2c212m2ξsn2c212m2ξ,
      where c0=c224c4(12m2)2, c2 > 0 and δ < 0. As m → 0, solution (69) transforms to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ1+εcos2c2ξsin2c2ξ,
      and as m → 1, solution (69) degenerates to solution (68). In solutions (63)(70), ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα.
    3. Set IIIa0=d2c4BA,a1=0,a1=(c2+2c4d2)2c42c4BA,ν=c2B,c0=c224c4
      u(x,y,z,t)=ε6c2(k12+k22+k32)δc2c4+2c4dtanhc22ξ2c4d+2c2c4tanhc22ξ,
      u(x,y,z,t)=ε6c2(k12+k22+k32)δc2c4+2c4dcothc22ξ2c4d+2c2c4cothc22ξ,
      where c2 < 0, c4 > 0 and δ < 0. Note that when d = 0 solution (71) converts to solution (64) whereas solution (72) turns to solution (56).
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ2c4dtanc22ξc2c42c2c4tanc22ξ+2c4d,
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ2c4dcotc22ξc2c42c2c4cotc22ξ+2c4d,
      where c2 > 0, c4 > 0 and δ < 0. In solutions (71)(74), ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα.
    4. Set IVa0=122BA(c2+2c4d2),a1=0,a1=d2BA(c2+2c4d2),ν=B2(c26c4d2),c0=c4d4
      u(x,y,z,t)=εm+126c2(k12+k22+k32)δ(m2+1)1msnc2m2+1ξ1+msnc2m2+1ξ,
      where c0=c22m2c4(m2+1)2, c2 < 0, δ > 0 and ξ=k1xαα+k2yαα+k3zαα+(m1)24m2(m2+1)c2k1(k12+k22+k32)tαα. As m → 1, solution (75) becomes
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ1itanhc22ξ1+itanhc22ξ.
      where ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα. Using the relation tanh(θ) = −i tan() solution (76) converts to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ1+tanc22ξ1tanc22ξ,
      where c2 > 0 and δ < 0.
      u(x,y,z,t)=εm21+m26c2(k12+k22+k32)δ(2m21)1mm21cnc22m21ξ1+mm21cnc22m21ξ,
      where c0=c22m2(m21)c4(2m21)2, c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+2m216mm212(2m21)c2k1(k12+k22+k32)tαα.
      u(x,y,z,t)=ε1m2+126c2(k12+k22+k32)δ(2m2)111m2dnc22m2ξ1+11m2dnc22m2ξ,
      where c0=c22(1m2)c4(2m2)2, c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+2m261m22(2m2)c2k1(k12+k22+k32)tαα.
      u(x,y,z,t)=ε3c2(1m2)(k12+k22+k32)δ(12m2)1+εcn2c212m2ξsn2c212m2ξ1+εcn2c212m2ξ+sn2c212m2ξ,
      where c0=c224c4(12m2)2, c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα(1+m2)c2k1(k12+k22+k32)tαα. As m → 0, solution (69) transforms to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δ1+εcos2c2ξsin2c2ξ1+εcos2c2ξ+sin2c2ξ,
      where ξ=k1xαα+k2yαα+k3zααc2k1(k12+k22+k32)tαα.
      u(x,y,z,t)=εd3c4(k12+k22+k32)δ12dsd2c4ξ,221+2dsd2c4ξ,22,
      where c2 = 0, c4 < 0 and δ > 0.
      u(x,y,z,t)=εd3c4(k12+k22+k32)δ1ns2dc4ξ,22cs2dc4ξ,221+ns2dc4ξ,22+cs2dc4ξ,22,
      where c2 = 0, c4 > 0 and δ < 0. In solutions (82) and (83), ξ=k1xαα+k2yαα+k3zαα3c4d2k1(k12+k22+k32)tαα.
    5. Set Va0=0,a1=2c4BA,a1=2c0BA,ν=B(6c0c4c2),d=0
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ(m2+1)1msn2c2m2+1ξsnc2m2+1ξ,
      where c0=c22m2c4(m2+1)2, c2 < 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+6mm21m2+1c2k1(k12+k22+k32)tαα. As m → 0, solution (84) converts to solution (44) while as m → 1, solution (84) degenerates to
      u(x,y,z,t)=ε3c2(k12+k22+k32)δsech2c22ξtanhc22ξ,
      where ξ=k1xαα+k2yαα+k3zαα+2c2k1(k12+k22+k32)tαα.
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ(2m21)m21mcn2c22m21ξcnc22m21ξ,
      where c0=c22m2(m21)c4(2m21)2, c2 > 0, δ > 0 and ξ=k1xαα+k2yαα+k3zαα+6mm212m2+12m21c2k1(k12+k22+k32)tαα. As m → 0, solution (86) converts to solution (43) while as m → 1, solution (86) degenerates to solution (41).
      u(x,y,z,t)=ε6c2(k12+k22+k32)δ(2m2)1m2dn2c22m2ξdnc22m2ξ,
      where c0=c22(1m2)c4(2m2)2, c2 > 0, δ > 0 and ξ=k1xαα+k2yαα+k3zαα+61m2+m222m2c2k1(k12+k22+k32)tαα. As m → 1, solution (87) reduces to solution (41).
      u(x,y,z,t)=2ε3c2(k12+k22+k32)δ(12m2)ns2c212m2ξ,
      where c0=c224c4(12m2)2, c2 > 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα+2(1+m2)12m2c2k1(k12+k22+k32)tαα. As m → 0, solution (88) reduces to
      u(x,y,z,t)=2ε3c2(k12+k22+k32)δcsc2c2ξ,
      where ξ=k1xαα+k2yαα+k3zαα+2c2k1(k12+k22+k32)tαα. As m → 1, solution (88) becomes
      u(x,y,z,t)=2ε3c2(k12+k22+k32)δcoth2c2ξ,
      where c2 < 0, δ < 0 and ξ=k1xαα+k2yαα+k3zαα4c2k1(k12+k22+k32)tαα.
      u(x,y,z,t)=2ε3c2(k12+k22+k32)δ(2m2)ds2c22m2ξ,
      where c0=c22m44c4(2m2)2, c2 < 0, δ > 0 and ξ=k1xαα+k2yαα+k3zαα+4m222m2c2k1(k12+k22+k32)tαα. As m → 0, solution (91) turns to solution (44) while as m → 1, solution (91) reduces to
      u(x,y,z,t)=2ε3c2(k12+k22+k32)δcsch2c2ξ,
      where c2 < 0, ξ=k1xαα+k2yαα+k3zαα+2c2k1(k12+k22+k32)tαα.
      u(x,y,z,t)=ε(c0c4)1/43(k12+k22+k32)δ1+2ids2(4c0c4)1/4ξ,22dn(4c0c4)1/4ξ,22,
      where c2 = 0, c0c4 < 0 and δ > 0.
      u(x,y,z,t)=ε(c0c4)1/46(k12+k22+k32)δ1+ns2(c0c4)1/4ξ,22+cs2(c0c4)1/4ξ,222ns2(c0c4)1/4ξ,22+cs2(c0c4)1/4ξ,22,
      where c2 = 0, c0c4 > 0 and δ < 0. In solutions (93) and (94), ξ=k1xαα+k2yαα+k3zαα+6c0c4k1(k12+k22+k32)tαα.

In what follows, we depict the dynamics of solitons and periodic waves in the model of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov equation to show a clear understanding of the physical properties of obtained results. The numerical results of some of the derived solutions are exhibited by selecting different values for the constants ε, δ, α, c2 and kl (l = 1, 2, 3). For example, the 3D and 2D plots of the bell-shaped solitary wave solution (41) are displayed in Figure 1 with ε = 1, k1 = 1, k2 = 1.2, k3 = 2.2, c2 = 1.5, δ = 2.5 when α = 0.95. Figure 2 shows the 3D and 2D plots of the kink-shaped solitary wave solution (45) for ε = −1, k1 = 0.2, k2 = 1.5, k3 = 0.25, c2 = 0.5, δ = −1.2 when α = 0.9. In Figure 3, the 3D and 2D plots of the singular soliton solution (50) are depicted for ε = 1, k1 = 0.5, k2 = 1.5, k3 = 0.3, c2 = 1.3, δ = −1 when α = 0.95. Further to this, the 3D and 2D plots of the singular periodic solution (70) are described in Figure 4 for ε = 1, k1 = 1, k2 = 0.5, k3 = 1.5, c2 = 2, δ = -0.5 when α = 1. Figure 5 shows the 3D and 2D plots of the singular periodic solution (89) for ε = 1, k1 = −1.5, k2 = 0.75, k3 = 1, c2 = 1.1, δ = −1.2 when α = 1. Figure 6 presents the 3D and 2D plots of the singular soliton solution (92) with ε = −1, k1 = 0.5, k2 = -0.5, k3 = 0.5, c2 = −1.2, δ = -1.5$ when α = 1.

Fig. 1
Fig. 1

The solitary wave solution (41) for k1 = 1, k2 = 1.2, k3 = 2.2, c2 = 1.5, δ = 2.5, α = 0.95.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00026

Fig. 2
Fig. 2

The kink wave solution (45) for k1 = 0.2, k2 = 1.5, k3 = 0.25, c2 = 0.5, δ = −1.2, α = 0.9.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00026

Fig. 3
Fig. 3

The singular soliton solution (50) for k1 = 0.5, k2 = 1.5, k3 = 0.3, c2 = 1.3, δ = −1, α = 0.95.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00026

Fig. 4
Fig. 4

The singular periodic solution (70) for k1 = 1, k2 = 0.5, k3 = 1.5, c2 = 2, δ = −0.5, α = 1.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00026

Fig. 5
Fig. 5

The singular periodic solution (89) for k1 = −1.5, k2 = 0.75, k3 = 1, c2 = 1.1, δ = −1.2, α = 1.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00026

Fig. 6
Fig. 6

The singular soliton solution (92) for k1 = 0.5, k2 = -0.5, k3 = 0.5, c2 = −1.2, δ = −1.5, α = 1.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00026

Remark 2

All of the results are calculated by using Maple, when t = 1 and z = 2 with the interval 0 < x, y ≤ 10.

5 Conclusions

In this study, we have investigated the exact analytic solutions to the (3+1)-dimensional space-time fractional mKdV–ZK equation. By means of conformable fractional derivative and wave transformation, the fractional mKdV–ZK equation is changed to an ODE. Then, the resulting ODE is solved by applying the variable separated ODE method. All obtained solutions are verified by utilising symbolic computation. To shed light on the behaviour of extracted results, some of derived solutions are displayed graphically. Moreover, it is found that the implemented method is a powerful mathematical tool for solving ODE and provides more exact solutions such as solitary, kink and periodic waves. Consequently, it can be applied to different physical models to generate various types of solutions.

Communicated by Juan Luis García Guirao

References

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    Kenneth S Miller and Bertram Ross. An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience New York 1993.

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    Igor Podlubny. Fractional differential equations volume 198. Academic Press California 1999.

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    A Anatolii Aleksandrovich Kilbas Hari Mohan Srivastava and Juan J Trujillo. Theory and applications of fractional differential equations volume 204. Elsevier Science Limited 2006.

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    Duan Zhao Xiao-Jun Yang and Hari M Srivastava. On the fractal heat transfer problems with local fractional calculus. Thermal Science 19(5):1867–1871 2015.

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    Xiao-Jun Yang JA Tenreiro Machado and HM Srivastava. A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach. Applied Mathematics and Computation 274:143–151 2016.

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    Elsayed ME Zayed Yasser A Amer and Reham MA Shohib. The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences 19:59–69 2016.

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    Yusuf Pandir and Ayse Yildirim. Analytical approach for the fractional differential equations by using the extended tanh method. Waves in Random and Complex Media 28(3):399–410 2018.

    • Crossref
    • Export Citation
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    S Zhang Q-A Zong D Liu and Q Gao. A generalized exp-function method for fractional Riccati differential equations. Commun.Frac.Calc. 1:48–51 2010.

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    Ozkan Guner Ahmet Bekir and Halis Bilgil. A note on exp-function method combined with complex transform method applied to fractional differential equations. Advances in Nonlinear Analysis 4(3):201–208 2015.

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    S Sahoo and S Saha Ray. Improved fractional sub-equation method for (3+ 1)-dimensional generalized fractional KdV–Zakharov–Kuznetsov equations. Computers & Mathematics with Applications 70(2):158–166 2015.

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    • Export Citation
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    Syed Tauseef Mohyud-Din Touqeer Nawaz Ehtsham Azhar and M Ali Akbar. Fractional sub-equation method to space–time fractional Calogero-Degasperis and potential Kadomtsev-Petviashvili equations. Journal of Taibah University for Science 11(2):258–263 2017.

    • Crossref
    • Export Citation
  • [12]

    Jalil Manafian and Mohammadreza Foroutan. Application of tan(ϕ(ξ)/2)-expansion method for the time-fractional Kuramoto–Sivashinsky equation. Optical and Quantum Electronics 49(8):272 2017.

    • Crossref
    • Export Citation
  • [13]

    Hadi Rezazadeh Jalil Manafian Farid Samsami Khodadad and Fakhroddin Nazari. Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved tan(12ψ$\begin{array}{}\displaystyle\frac{1}{2}\psi\end{array}$(ξ))-expansion method. Optical and Quantum Electronics 50(3):121 2018.

    • Crossref
    • Export Citation
  • [14]

    Zheng Bin. (G′/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Communications in Theoretical Physics 58(5):623 2012.

    • Crossref
    • Export Citation
  • [15]

    Ahmet Bekir and Özkan Güner. Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method. Chinese Physics B 22(11):110202 2013.

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    Hasan Bulut Haci Mehmet Baskonus and Yusuf Pandir. The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation. In Abstract and Applied Analysis volume 2013. Hindawi 2013.

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    Meryem Odabasi and Emine Misirli. On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences 41(3):904–911 2018.

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    • Export Citation
  • [18]

    Hadi Rezazadeh Hira Tariq Mostafa Eslami Mohammad Mirzazadeh and Qin Zhou. New exact solutions of nonlinear conformable time-fractional Phi-4 equation. Chinese Journal of Physics 56(6):2805–2816 2018.

    • Crossref
    • Export Citation
  • [19]

    Hadi Rezazadeh. New solitons solutions of the complex Ginzburg-Landau equation with kerr law nonlinearity. Optik 167:218–227 2018.

    • Crossref
    • Export Citation
  • [20]

    Hasan Bulut Tukur Abdulkadir Sulaiman Haci Mehmet Baskonus Hadi Rezazadeh Mostafa Eslami and Mohammad Mirzazadeh. Optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation. Optik 172:20–27 2018.

    • Crossref
    • Export Citation
  • [21]

    Hadi Rezazadeh Mohammad Mirzazadeh Seyed Mehdi Mirhosseini-Alizamini Ahmad Neirameh Mostafa Eslami and Qin Zhou. Optical solitons of Lakshmanan–Porsezian–Daniel model with a couple of nonlinearities. Optik 164:414–423 2018.

    • Crossref
    • Export Citation
  • [22]

    MS Osman Alper Korkmaz Hadi Rezazadeh Mohammad Mirzazadeh Mostafa Eslami and Qin Zhou. The unified method for conformable time fractional Schrdinger equation with perturbation terms. Chinese Journal of Physics 2018.

  • [23]

    Hasibun Naher Farah Aini Abdullah and M Ali Akbar. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method. Journal of Applied Mathematics 2012 2012.

  • [24]

    Hasibun Naher Farah Aini Abdullah and M Ali Akbar. Generalized and improved (G′/G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PloS one 8(5):e64618 2013.

    • Crossref
    • PubMed
    • Export Citation
  • [25]

    Md Hamidul Islam Kamruzzaman Khan M Ali Akbar and Md Abdus Salam. Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation. SpringerPlus 3(1):105 2014.

    • Crossref
    • PubMed
    • Export Citation
  • [26]

    RL Mace and MA Hellberg. The Korteweg–de Vries–Zakharov–Kuznetsov equation for electron-acoustic waves. Physics of Plasmas 8(6):2649–2656 2001.

    • Crossref
    • Export Citation
  • [27]

    Ozkan Guner Esin Aksoy Ahmet Bekir and Adem C Cevikel. Different methods for (3+1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation. Computers & Mathematics with Applications 71(6):1259–1269 2016.

    • Crossref
    • Export Citation
  • [28]

    Ozkan Guner. New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods. Optical and Quantum Electronics 50(1):38 2018.

    • Crossref
    • Export Citation
  • [29]

    Mohamed AE Herzallah. Comments on “different methods for (3+1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation” [comput. math. appl. 71 (6)(2016) 1259–1269]. Computers & Mathematics with Applications 77(1):66–68 2019.

    • Crossref
    • Export Citation
  • [30]

    Emmanuel Yomba. The extended Fan’s sub-equation method and its application to KdV–MKdV BKK and variant Boussinesqequations. Physics Letters A 336(6):463–476 2005.

    • Crossref
    • Export Citation
  • [31]

    Sirendaoreji. Variable separated ODE method–A powerful tool for testing traveling wave solutions of nonlinear equations. arXiv e-prints page arXiv:1811.05406 November 2018.

  • [32]

    Roshdi Khalil Mohammed Al Horani Abdelrahman Yousef and Mohammad Sababheh. A new definition of fractional derivative. Journal of Computational and Applied Mathematics 264:65–70 2014.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Kenneth S Miller and Bertram Ross. An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience New York 1993.

  • [2]

    Igor Podlubny. Fractional differential equations volume 198. Academic Press California 1999.

  • [3]

    A Anatolii Aleksandrovich Kilbas Hari Mohan Srivastava and Juan J Trujillo. Theory and applications of fractional differential equations volume 204. Elsevier Science Limited 2006.

  • [4]

    Duan Zhao Xiao-Jun Yang and Hari M Srivastava. On the fractal heat transfer problems with local fractional calculus. Thermal Science 19(5):1867–1871 2015.

    • Crossref
    • Export Citation
  • [5]

    Xiao-Jun Yang JA Tenreiro Machado and HM Srivastava. A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach. Applied Mathematics and Computation 274:143–151 2016.

    • Crossref
    • Export Citation
  • [6]

    Elsayed ME Zayed Yasser A Amer and Reham MA Shohib. The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences 19:59–69 2016.

  • [7]

    Yusuf Pandir and Ayse Yildirim. Analytical approach for the fractional differential equations by using the extended tanh method. Waves in Random and Complex Media 28(3):399–410 2018.

    • Crossref
    • Export Citation
  • [8]

    S Zhang Q-A Zong D Liu and Q Gao. A generalized exp-function method for fractional Riccati differential equations. Commun.Frac.Calc. 1:48–51 2010.

  • [9]

    Ozkan Guner Ahmet Bekir and Halis Bilgil. A note on exp-function method combined with complex transform method applied to fractional differential equations. Advances in Nonlinear Analysis 4(3):201–208 2015.

  • [10]

    S Sahoo and S Saha Ray. Improved fractional sub-equation method for (3+ 1)-dimensional generalized fractional KdV–Zakharov–Kuznetsov equations. Computers & Mathematics with Applications 70(2):158–166 2015.

    • Crossref
    • Export Citation
  • [11]

    Syed Tauseef Mohyud-Din Touqeer Nawaz Ehtsham Azhar and M Ali Akbar. Fractional sub-equation method to space–time fractional Calogero-Degasperis and potential Kadomtsev-Petviashvili equations. Journal of Taibah University for Science 11(2):258–263 2017.

    • Crossref
    • Export Citation
  • [12]

    Jalil Manafian and Mohammadreza Foroutan. Application of tan(ϕ(ξ)/2)-expansion method for the time-fractional Kuramoto–Sivashinsky equation. Optical and Quantum Electronics 49(8):272 2017.

    • Crossref
    • Export Citation
  • [13]

    Hadi Rezazadeh Jalil Manafian Farid Samsami Khodadad and Fakhroddin Nazari. Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved tan(12ψ$\begin{array}{}\displaystyle\frac{1}{2}\psi\end{array}$(ξ))-expansion method. Optical and Quantum Electronics 50(3):121 2018.

    • Crossref
    • Export Citation
  • [14]

    Zheng Bin. (G′/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Communications in Theoretical Physics 58(5):623 2012.

    • Crossref
    • Export Citation
  • [15]

    Ahmet Bekir and Özkan Güner. Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method. Chinese Physics B 22(11):110202 2013.

  • [16]

    Hasan Bulut Haci Mehmet Baskonus and Yusuf Pandir. The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation. In Abstract and Applied Analysis volume 2013. Hindawi 2013.

  • [17]

    Meryem Odabasi and Emine Misirli. On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Mathematical Methods in the Applied Sciences 41(3):904–911 2018.

    • Crossref
    • Export Citation
  • [18]

    Hadi Rezazadeh Hira Tariq Mostafa Eslami Mohammad Mirzazadeh and Qin Zhou. New exact solutions of nonlinear conformable time-fractional Phi-4 equation. Chinese Journal of Physics 56(6):2805–2816 2018.

    • Crossref
    • Export Citation
  • [19]

    Hadi Rezazadeh. New solitons solutions of the complex Ginzburg-Landau equation with kerr law nonlinearity. Optik 167:218–227 2018.

    • Crossref
    • Export Citation
  • [20]

    Hasan Bulut Tukur Abdulkadir Sulaiman Haci Mehmet Baskonus Hadi Rezazadeh Mostafa Eslami and Mohammad Mirzazadeh. Optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation. Optik 172:20–27 2018.

    • Crossref
    • Export Citation
  • [21]

    Hadi Rezazadeh Mohammad Mirzazadeh Seyed Mehdi Mirhosseini-Alizamini Ahmad Neirameh Mostafa Eslami and Qin Zhou. Optical solitons of Lakshmanan–Porsezian–Daniel model with a couple of nonlinearities. Optik 164:414–423 2018.

    • Crossref
    • Export Citation
  • [22]

    MS Osman Alper Korkmaz Hadi Rezazadeh Mohammad Mirzazadeh Mostafa Eslami and Qin Zhou. The unified method for conformable time fractional Schrdinger equation with perturbation terms. Chinese Journal of Physics 2018.

  • [23]

    Hasibun Naher Farah Aini Abdullah and M Ali Akbar. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method. Journal of Applied Mathematics 2012 2012.

  • [24]

    Hasibun Naher Farah Aini Abdullah and M Ali Akbar. Generalized and improved (G′/G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. PloS one 8(5):e64618 2013.

    • Crossref
    • PubMed
    • Export Citation
  • [25]

    Md Hamidul Islam Kamruzzaman Khan M Ali Akbar and Md Abdus Salam. Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation. SpringerPlus 3(1):105 2014.

    • Crossref
    • PubMed
    • Export Citation
  • [26]

    RL Mace and MA Hellberg. The Korteweg–de Vries–Zakharov–Kuznetsov equation for electron-acoustic waves. Physics of Plasmas 8(6):2649–2656 2001.

    • Crossref
    • Export Citation
  • [27]

    Ozkan Guner Esin Aksoy Ahmet Bekir and Adem C Cevikel. Different methods for (3+1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation. Computers & Mathematics with Applications 71(6):1259–1269 2016.

    • Crossref
    • Export Citation
  • [28]

    Ozkan Guner. New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods. Optical and Quantum Electronics 50(1):38 2018.

    • Crossref
    • Export Citation
  • [29]

    Mohamed AE Herzallah. Comments on “different methods for (3+1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation” [comput. math. appl. 71 (6)(2016) 1259–1269]. Computers & Mathematics with Applications 77(1):66–68 2019.

    • Crossref
    • Export Citation
  • [30]

    Emmanuel Yomba. The extended Fan’s sub-equation method and its application to KdV–MKdV BKK and variant Boussinesqequations. Physics Letters A 336(6):463–476 2005.

    • Crossref
    • Export Citation
  • [31]

    Sirendaoreji. Variable separated ODE method–A powerful tool for testing traveling wave solutions of nonlinear equations. arXiv e-prints page arXiv:1811.05406 November 2018.

  • [32]

    Roshdi Khalil Mohammed Al Horani Abdelrahman Yousef and Mohammad Sababheh. A new definition of fractional derivative. Journal of Computational and Applied Mathematics 264:65–70 2014.

    • Crossref
    • Export Citation
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Journal information
Figures
  • View in gallery

    The solitary wave solution (41) for k1 = 1, k2 = 1.2, k3 = 2.2, c2 = 1.5, δ = 2.5, α = 0.95.

  • View in gallery

    The kink wave solution (45) for k1 = 0.2, k2 = 1.5, k3 = 0.25, c2 = 0.5, δ = −1.2, α = 0.9.

  • View in gallery

    The singular soliton solution (50) for k1 = 0.5, k2 = 1.5, k3 = 0.3, c2 = 1.3, δ = −1, α = 0.95.

  • View in gallery

    The singular periodic solution (70) for k1 = 1, k2 = 0.5, k3 = 1.5, c2 = 2, δ = −0.5, α = 1.

  • View in gallery

    The singular periodic solution (89) for k1 = −1.5, k2 = 0.75, k3 = 1, c2 = 1.1, δ = −1.2, α = 1.

  • View in gallery

    The singular soliton solution (92) for k1 = 0.5, k2 = -0.5, k3 = 0.5, c2 = −1.2, δ = −1.5, α = 1.

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